Jarvis has had a long association with the BBC, particularly BBC Radio 4. In the 1980s Michael Frayn's columns for the Guardian and the Observer, described by some as models of the comic essay, were adapted and performed in many voices for BBC Radio 4 by Jarvis. He performs regularly in radio dramas and readings, both comic and serious. In David Mamet's Mind Your Pantheon he played the actor Strabo. He is probably best known[citation needed] for his long series of readings of Richmal Crompton's Just William stories, which show his characteristic and flexible reading voices. He has also narrated the Billy Bunter series, by Frank Richards. As a result of this extensive work Jarvis is satirised by the radio show Dead Ringers by Mark Perry, highlighting his seeming ubiquity on Radio 4 programmes and as a guest in Dictionary Corner on Countdown.

Jarvis has also lent his voice to audiobooks on Sir P.G. Wodehouse's works and has won the Audie Award for the same. He is the narrator of the 2011 audiobook of The Selfish Giant by Oscar Wilde. Further work in 2011 includes the audiobook for the Wilbur Smith novel 'The Leopard Hunts in Darkness'. In film, Jarvis appeared in Disney's 2012 film Wreck-It Ralph as Saitine, one of the video game villains who attends the Bad Anon self-help groups with Ralph (John C. Reilly).

He married Rosalind Ayres on 23 November 1974 in Ealing and has two sons by a previous marriage. He met Ayres when playing in Hamlet together, where she played Ophelia. Together with his wife, Jarvis runs the radio production company "Jarvis & Ayres Productions",[2] frequently used by Radio 4.